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Parallel reductions in \(\lambda\)-calculus. (English) Zbl 0661.03008
The notion of parallel reduction is extracted from the Tait-Martin-Löf proof of the Church-Rosser theorem (for \(\beta\)-reduction). We define parallel \(\beta\)-, \(\eta\)- and \(\beta\) \(\eta\)-reduction by induction, and use them to give simple proofs of some fundamental theorems in \(\lambda\)- calculus; the normal reduction theorem for \(\beta\)-reduction, that for \(\beta\) \(\eta\)-reduction, the postponement theorem of \(\eta\)-reduction (in \(\beta\) \(\eta\)-reduction), and some others.

MSC:
03B40 Combinatory logic and lambda calculus
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References:
[1] Barendregt, H.P., ()
[2] Klop, J.W., ()
[3] Levy, J.J., An algebraic interpretation of the λ-β-K calculus and a labelled λ-calculus, Springer lec. notes comp., 37, 147-165, (1975)
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